2005 Fiscal Year Final Research Report Summary
Bergman zeta function and index theorems of complex domains
| Project/Area Number |
15340040
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | The University of Tokyo |
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Principal Investigator |
HIRACHI Kengo The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理学研究科, 助教授 (60218790)
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| Co-Investigator(Kenkyū-buntansha) |
KOMATSU Gen Osaka University, Graduate School of Sciences, Associate Professor, 大学院・理学研究科, 助教授 (60108446)
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| Project Period (FY) |
2003 – 2005
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| Keywords | Bergman kernel / Parabolic invariant theory / CR geometry / strictly pseudoconvex domain / conformal geometry |
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| Research Abstract |
The Bergman-zeta function is a meromorphic function of one complex variable that is defined by the analytic continuation of the integral of weighted Bergman kernel on the diagonal ; here the integral is considered as a function of a parameter s with which we define the weight r^s for a domain r>0. We fist showed that the residues of the Bergman-zeta function contain a biholomorphic invariant and proved that the invariant is given by the integral of a local pseudo-hermitian invariant P, which is defined as the log term of the Szego kernel. In 2 dimensions, we also showed that P agrees with the CR Q-curvature, which is defined via a CR invariant differential operator, while for higher dimensions, the relation between P and CR Q-curvature was difficult to analyze. We thus also studied the Q-curvature in conformal geometry, which is the area where Q-curvature was originally introduced. The definition of the Q-curvature was not so clear as it is based on an argument using analytic continuation in dimension. We thus gave (with Prof.Fefferman) an explicit formula of the Q-curvature in terms of the ambient metric. We also showed (with Prof.Graham) that the variation of the integral of the Q-curvature in a deformation of conformal structure is given by the Fefferman-Graham obstruction tensor. These result can be translated to the case of CR Q-curvature and give its expression in terms of the ambient metric of the CR structure. This argument, together with parabolic invariant theory, gives, in 2-dimensions, a simple proof of the fact that P agrees with CR Q-curvature, and for higher dimensions, a way to construct several pseudo-hermitian invariant that has invariance property similar to CR Q-curvature; we hope to purse this approach and write the difference between P and CR Q-curvature.
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Research Products
(10 results)
